A mathematical model with nonlinear relapse: conditions for a forward-backward bifurcation

We constructed a Susceptible-Addicted-Reformed model and explored the dynamics of nonlinear relapse in the Reformed population. The transition from susceptible considered at-risk is modeled using a strictly decreasing general function, mimicking an influential factor that reduces the flow into the addicted class. The basic reproductive number is computed, which determines the local asymptotically stability of the addicted-free equilibrium. Conditions for a forward-backward bifurcation were established using the basic reproductive number and other threshold quantities. A stochastic version of the model is presented, and some numerical examples are shown. Results showed that the influence of the temporarily reformed individuals is highly sensitive to the initial addicted population.

Automated summary

A Susceptible-Addicted-Reformed model was constructed to examine the dynamics of nonlinear relapse in the Reformed population. The transition from susceptible to at-risk was modeled using a strictly decreasing general function as an influential factor. The basic reproductive number was computed to determine the stability of the addicted-free equilibrium, and conditions for a forward-backward bifurcation were established.